Numbers and Sets | Part IA, 2004

Show that the set of all subsets of N\mathbb{N} is uncountable, and that the set of all finite subsets of N\mathbb{N} is countable.

Let XX be the set of all bijections from N\mathbb{N} to N\mathbb{N}, and let YXY \subset X be the set

Y={fX for all but finitely many nN,f(n)=n}Y=\{f \in X \mid \text { for all but finitely many } n \in \mathbb{N}, f(n)=n\}

Show that XX is uncountable, but that YY is countable.

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